Geology Reference
In-Depth Information
However,
v
is just the rate of decrease of
d
, so we can write
∂
d
∂
t
=−
gRρ
μ
d,
(A.7)
where the differentials have been written with the ∂ symbol to distinguish them from the
variable
d
, and
ρ
ρ
w
).
The collection of constants on the right-hand side of Eq. (A.7) has the dimensions of
inverse time, so let's define a time constant as
τ
=
(
ρ
m
−
=
μ/gRρ.
(A.8)
Then Eq. (A.7) can be written as
∂
d
∂
t
=−
d
τ
.
(A.9)
This has the same form as Eq. (A.2). We can identify
d
with
y
and
a
with
−
1
/τ
and take
the solution from Eq. (A.5):
d
=
d
0
exp (
−
t/τ
)
.
(A.10)
Thus we derive that the depth of the post-glacial depression decreases exponentially
with time. As Figure 4.2 shows, an exponential decline closely fits the observed change in
relative sea level in Fennoscandia, and the best-fit exponential has a decay timescale of
4.6 kyr. If we identify this with
τ
in Eq. (A.8), then we get another expression for mantle
viscosity:
μ
=
τgRρ.
(A.11)
10 m/s
2
,
R
2300 kg/m
3
, this yields
μ
10
21
Pa s.
Using
g
=
=
1000 km and
ρ
=
=
3.3
×
A.3 Rayleigh-Taylor instability
In Section 7.3.2, the instability of a fluid layer is analysed by considering a bulge of height
h
on an interface between low-density fluid below and higher-density fluid above.
Equation (7.13) is given for the velocity,
v
, of the top of the bulge:
v
=
(
gρw/μ
)
h.
The form of this relationship is very similar to the post-glacial relationship in the
preceding section, except here the bulge is growing, so
v
=
d
h/
d
t
and
∂
h
∂
t
=
gwρ
μ
h
τ
.
h
=
(A.12)
This has the same form as Eq. (A.2), so we can identify
h
with
y
and
a
with 1
/τ
and write
the solution as
h
0
exp(
t/τ
)
,
(A.13)
which is the same as Eq. (7.14). This describes exponential growth of the height of the
bulge. In other words, the bulge is unstable and grows with a timescale of
τ
.
h
=
